**Decreasing Order of Stability of Vinyl, Allyl, and Tertiary Free Radicals**

Free radicals are highly reactive species with an unpaired electron, making them unstable. The stability of a free radical is determined by various factors, including electron delocalization, hyperconjugation, and inductive effects. In the case of vinyl, allyl, and tertiary free radicals, their stability can be compared based on these factors.

**1. Vinyl Free Radical**

A vinyl free radical is formed by the removal of a hydrogen atom from a carbon atom in a vinyl group (CH2=CH-). The stability of a vinyl free radical is relatively low due to the absence of any stabilizing factors.

- Lack of hyperconjugation: The presence of a double bond restricts the rotation of the sigma bonds, limiting the hyperconjugation effect. This leads to a weaker stabilization of the radical.
- Lack of electron delocalization: The unpaired electron in the vinyl free radical is localized on the carbon atom, resulting in a high electron density and instability.

**2. Allyl Free Radical**

An allyl free radical is formed by the removal of a hydrogen atom from the central carbon atom in an allyl group (CH2=CH-CH2-). The allyl free radical is more stable than the vinyl free radical due to the presence of additional stabilizing factors.

- Hyperconjugation: The allyl group contains two sigma bonds adjacent to the carbon with the unpaired electron. These sigma bonds can donate electron density through hyperconjugation, stabilizing the radical.
- Electron delocalization: The unpaired electron in the allyl free radical is delocalized over the three carbon atoms, allowing for better dispersion of the electron density. This delocalization increases the stability of the radical.

**3. Tertiary Free Radical**

A tertiary free radical is formed by the removal of a hydrogen atom from a tertiary carbon atom (R3C-). Tertiary free radicals are the most stable among the three discussed radicals due to the presence of multiple stabilizing factors.

- Hyperconjugation: Tertiary carbon atoms have three alkyl groups attached to them, providing a large number of sigma bonds for hyperconjugation. This results in significant electron delocalization and stabilization of the radical.
- Inductive effect: The alkyl groups attached to the tertiary carbon atom exert an inductive effect, donating electron density towards the carbon with the unpaired electron. This further stabilizes the tertiary free radical.

**Conclusion**

In conclusion, the decreasing order of stability for vinyl, allyl, and tertiary free radicals is as follows:

1. Tertiary free radical (most stable)
2. Allyl free radical
3. Vinyl free radical (least stable)